On density of subgraphs of Cartesian products

Abstract

In this paper, we extend two classical results about the density of subgraphs of hypercubes to subgraphs (Formula presented.) of Cartesian products (Formula presented.) of arbitrary connected graphs. Namely, we show that (Formula presented.), where (Formula presented.) is the maximum ratio (Formula presented.) taken over all subgraphs (Formula presented.) of (Formula presented.). We introduce the notions of VC-dimension (Formula presented.) and VC-density (Formula presented.) of a subgraph (Formula presented.) of a Cartesian product (Formula presented.), generalizing the classical Vapnik-Chervonenkis dimension of set-families (viewed as subgraphs of hypercubes). We prove that if (Formula presented.) belong to the class (Formula presented.) of all finite connected graphs not containing a given graph (Formula presented.) as a minor, then for any subgraph (Formula presented.) of (Formula presented.) the sharper inequality (Formula presented.) holds, where (Formula presented.) is the supremum of the densities of the graphs from (Formula presented.). We refine and sharpen these two results to several specific graph classes. We also derive upper bounds (some of them polylogarithmic) for the size of adjacency labeling schemes of subgraphs of Cartesian products.